@article{1216, abstract = {A framework fo r extracting features in 2D transient flows, based on the acceleration field to ensure Galilean invariance is proposed in this paper. The minima of the acceleration magnitude (a superset of acceleration zeros) are extracted and discriminated into vortices and saddle points, based on the spectral properties of the velocity Jacobian. The extraction of topological features is performed with purely combinatorial algorithms from discrete computational topology. The feature points are prioritized with persistence, as a physically meaningful importance measure. These feature points are tracked in time with a robust algorithm for tracking features. Thus, a space-time hierarchy of the minima is built and vortex merging events are detected. We apply the acceleration feature extraction strategy to three two-dimensional shear flows: (1) an incompressible periodic cylinder wake, (2) an incompressible planar mixing layer and (3) a weakly compressible planar jet. The vortex-like acceleration feature points are shown to be well aligned with acceleration zeros, maxima of the vorticity magnitude, minima of the pressure field and minima of λ2.}, author = {Kasten, Jens and Reininghaus, Jan and Hotz, Ingrid and Hege, Hans and Noack, Bernd and Daviller, Guillaume and Morzyński, Marek}, journal = {Archives of Mechanics}, number = {1}, pages = {55 -- 80}, publisher = {Polish Academy of Sciences Publishing House}, title = {{Acceleration feature points of unsteady shear flows}}, volume = {68}, year = {2016}, } @inproceedings{1483, abstract = {Topological data analysis offers a rich source of valuable information to study vision problems. Yet, so far we lack a theoretically sound connection to popular kernel-based learning techniques, such as kernel SVMs or kernel PCA. In this work, we establish such a connection by designing a multi-scale kernel for persistence diagrams, a stable summary representation of topological features in data. We show that this kernel is positive definite and prove its stability with respect to the 1-Wasserstein distance. Experiments on two benchmark datasets for 3D shape classification/retrieval and texture recognition show considerable performance gains of the proposed method compared to an alternative approach that is based on the recently introduced persistence landscapes.}, author = {Reininghaus, Jan and Huber, Stefan and Bauer, Ulrich and Kwitt, Roland}, location = {Boston, MA, USA}, pages = {4741 -- 4748}, publisher = {IEEE}, title = {{A stable multi-scale kernel for topological machine learning}}, doi = {10.1109/CVPR.2015.7299106}, year = {2015}, } @inbook{1531, abstract = {The Heat Kernel Signature (HKS) is a scalar quantity which is derived from the heat kernel of a given shape. Due to its robustness, isometry invariance, and multiscale nature, it has been successfully applied in many geometric applications. From a more general point of view, the HKS can be considered as a descriptor of the metric of a Riemannian manifold. Given a symmetric positive definite tensor field we may interpret it as the metric of some Riemannian manifold and thereby apply the HKS to visualize and analyze the given tensor data. In this paper, we propose a generalization of this approach that enables the treatment of indefinite tensor fields, like the stress tensor, by interpreting them as a generator of a positive definite tensor field. To investigate the usefulness of this approach we consider the stress tensor from the two-point-load model example and from a mechanical work piece.}, author = {Zobel, Valentin and Reininghaus, Jan and Hotz, Ingrid}, booktitle = {Visualization and Processing of Higher Order Descriptors for Multi-Valued Data}, editor = {Hotz, Ingrid and Schultz, Thomas}, isbn = {978-3-319-15089-5}, pages = {257 -- 267}, publisher = {Springer}, title = {{Visualizing symmetric indefinite 2D tensor fields using The Heat Kernel Signature}}, doi = {10.1007/978-3-319-15090-1_13}, volume = {40}, year = {2015}, } @inbook{10893, abstract = {Saddle periodic orbits are an essential and stable part of the topological skeleton of a 3D vector field. Nevertheless, there is currently no efficient algorithm to robustly extract these features. In this chapter, we present a novel technique to extract saddle periodic orbits. Exploiting the analytic properties of such an orbit, we propose a scalar measure based on the finite-time Lyapunov exponent (FTLE) that indicates its presence. Using persistent homology, we can then extract the robust cycles of this field. These cycles thereby represent the saddle periodic orbits of the given vector field. We discuss the different existing FTLE approximation schemes regarding their applicability to this specific problem and propose an adapted version of FTLE called Normalized Velocity Separation. Finally, we evaluate our method using simple analytic vector field data.}, author = {Kasten, Jens and Reininghaus, Jan and Reich, Wieland and Scheuermann, Gerik}, booktitle = {Topological Methods in Data Analysis and Visualization III }, editor = {Bremer, Peer-Timo and Hotz, Ingrid and Pascucci, Valerio and Peikert, Ronald}, isbn = {9783319040981}, issn = {2197-666X}, pages = {55--69}, publisher = {Springer}, title = {{Toward the extraction of saddle periodic orbits}}, doi = {10.1007/978-3-319-04099-8_4}, volume = {1}, year = {2014}, } @article{1930, abstract = {(Figure Presented) Data acquisition, numerical inaccuracies, and sampling often introduce noise in measurements and simulations. Removing this noise is often necessary for efficient analysis and visualization of this data, yet many denoising techniques change the minima and maxima of a scalar field. For example, the extrema can appear or disappear, spatially move, and change their value. This can lead to wrong interpretations of the data, e.g., when the maximum temperature over an area is falsely reported being a few degrees cooler because the denoising method is unaware of these features. Recently, a topological denoising technique based on a global energy optimization was proposed, which allows the topology-controlled denoising of 2D scalar fields. While this method preserves the minima and maxima, it is constrained by the size of the data. We extend this work to large 2D data and medium-sized 3D data by introducing a novel domain decomposition approach. It allows processing small patches of the domain independently while still avoiding the introduction of new critical points. Furthermore, we propose an iterative refinement of the solution, which decreases the optimization energy compared to the previous approach and therefore gives smoother results that are closer to the input. We illustrate our technique on synthetic and real-world 2D and 3D data sets that highlight potential applications.}, author = {Günther, David and Jacobson, Alec and Reininghaus, Jan and Seidel, Hans and Sorkine Hornung, Olga and Weinkauf, Tino}, journal = {IEEE Transactions on Visualization and Computer Graphics}, number = {12}, pages = {2585 -- 2594}, publisher = {IEEE}, title = {{Fast and memory-efficient topological denoising of 2D and 3D scalar fields}}, doi = {10.1109/TVCG.2014.2346432}, volume = {20}, year = {2014}, } @inproceedings{2043, abstract = {Persistent homology is a popular and powerful tool for capturing topological features of data. Advances in algorithms for computing persistent homology have reduced the computation time drastically – as long as the algorithm does not exhaust the available memory. Following up on a recently presented parallel method for persistence computation on shared memory systems [1], we demonstrate that a simple adaption of the standard reduction algorithm leads to a variant for distributed systems. Our algorithmic design ensures that the data is distributed over the nodes without redundancy; this permits the computation of much larger instances than on a single machine. Moreover, we observe that the parallelism at least compensates for the overhead caused by communication between nodes, and often even speeds up the computation compared to sequential and even parallel shared memory algorithms. In our experiments, we were able to compute the persistent homology of filtrations with more than a billion (109) elements within seconds on a cluster with 32 nodes using less than 6GB of memory per node.}, author = {Bauer, Ulrich and Kerber, Michael and Reininghaus, Jan}, booktitle = {Proceedings of the Workshop on Algorithm Engineering and Experiments}, editor = { McGeoch, Catherine and Meyer, Ulrich}, location = {Portland, USA}, pages = {31 -- 38}, publisher = {Society of Industrial and Applied Mathematics}, title = {{Distributed computation of persistent homology}}, doi = {10.1137/1.9781611973198.4}, year = {2014}, } @inbook{2044, abstract = {We present a parallel algorithm for computing the persistent homology of a filtered chain complex. Our approach differs from the commonly used reduction algorithm by first computing persistence pairs within local chunks, then simplifying the unpaired columns, and finally applying standard reduction on the simplified matrix. The approach generalizes a technique by Günther et al., which uses discrete Morse Theory to compute persistence; we derive the same worst-case complexity bound in a more general context. The algorithm employs several practical optimization techniques, which are of independent interest. Our sequential implementation of the algorithm is competitive with state-of-the-art methods, and we further improve the performance through parallel computation.}, author = {Bauer, Ulrich and Kerber, Michael and Reininghaus, Jan}, booktitle = {Topological Methods in Data Analysis and Visualization III}, editor = {Bremer, Peer-Timo and Hotz, Ingrid and Pascucci, Valerio and Peikert, Ronald}, pages = {103 -- 117}, publisher = {Springer}, title = {{Clear and Compress: Computing Persistent Homology in Chunks}}, doi = {10.1007/978-3-319-04099-8_7}, year = {2014}, } @inproceedings{10886, abstract = {We propose a method for visualizing two-dimensional symmetric positive definite tensor fields using the Heat Kernel Signature (HKS). The HKS is derived from the heat kernel and was originally introduced as an isometry invariant shape signature. Each positive definite tensor field defines a Riemannian manifold by considering the tensor field as a Riemannian metric. On this Riemmanian manifold we can apply the definition of the HKS. The resulting scalar quantity is used for the visualization of tensor fields. The HKS is closely related to the Gaussian curvature of the Riemannian manifold and the time parameter of the heat kernel allows a multiscale analysis in a natural way. In this way, the HKS represents field related scale space properties, enabling a level of detail analysis of tensor fields. This makes the HKS an interesting new scalar quantity for tensor fields, which differs significantly from usual tensor invariants like the trace or the determinant. A method for visualization and a numerical realization of the HKS for tensor fields is proposed in this chapter. To validate the approach we apply it to some illustrating simple examples as isolated critical points and to a medical diffusion tensor data set.}, author = {Zobel, Valentin and Reininghaus, Jan and Hotz, Ingrid}, booktitle = {Topological Methods in Data Analysis and Visualization III }, isbn = {9783319040981}, issn = {2197-666X}, pages = {249--262}, publisher = {Springer}, title = {{Visualization of two-dimensional symmetric positive definite tensor fields using the heat kernel signature}}, doi = {10.1007/978-3-319-04099-8_16}, year = {2014}, } @inbook{10817, abstract = {The Morse-Smale complex can be either explicitly or implicitly represented. Depending on the type of representation, the simplification of the Morse-Smale complex works differently. In the explicit representation, the Morse-Smale complex is directly simplified by explicitly reconnecting the critical points during the simplification. In the implicit representation, on the other hand, the Morse-Smale complex is given by a combinatorial gradient field. In this setting, the simplification changes the combinatorial flow, which yields an indirect simplification of the Morse-Smale complex. The topological complexity of the Morse-Smale complex is reduced in both representations. However, the simplifications generally yield different results. In this chapter, we emphasize properties of the two representations that cause these differences. We also provide a complexity analysis of the two schemes with respect to running time and memory consumption.}, author = {Günther, David and Reininghaus, Jan and Seidel, Hans-Peter and Weinkauf, Tino}, booktitle = {Topological Methods in Data Analysis and Visualization III.}, editor = {Bremer, Peer-Timo and Hotz, Ingrid and Pascucci, Valerio and Peikert, Ronald}, isbn = {9783319040981}, issn = {2197-666X}, pages = {135--150}, publisher = {Springer Nature}, title = {{Notes on the simplification of the Morse-Smale complex}}, doi = {10.1007/978-3-319-04099-8_9}, year = {2014}, } @inproceedings{10894, abstract = {PHAT is a C++ library for the computation of persistent homology by matrix reduction. We aim for a simple generic design that decouples algorithms from data structures without sacrificing efficiency or user-friendliness. This makes PHAT a versatile platform for experimenting with algorithmic ideas and comparing them to state of the art implementations.}, author = {Bauer, Ulrich and Kerber, Michael and Reininghaus, Jan and Wagner, Hubert}, booktitle = {ICMS 2014: International Congress on Mathematical Software}, isbn = {9783662441985}, issn = {1611-3349}, location = {Seoul, South Korea}, pages = {137--143}, publisher = {Springer Berlin Heidelberg}, title = {{PHAT – Persistent Homology Algorithms Toolbox}}, doi = {10.1007/978-3-662-44199-2_24}, volume = {8592}, year = {2014}, }